Introduction to Python Day 6

Verjinia Metodieva and Daniel Parthier

2025-04-01

“Visualization is a method of computing. It transforms the symbolic into the geometric, enabling researchers to observe their simulations and computations. Visualization offers a method for seeing the unseen. It enriches the process of scientific discovery and fosters profound and unexpected insights. In many fields it is already revolutionizing the way scientists do science.

~ McCormick, B.H., T.A. DeFanti, M.D. Brown, Visualization in Scientific Computing, Computer Graphics Vol. 21.6, November 1987

“Sometimes the most effective way to describe, explore, and summarize a set of numbers - even a very large set - is to look at those numbers”

~ The Visual Display of Quantitative Information, Edward Tufte, 1983

  • audience, message, caption, choice of viz, no misleading plotting

Matplotlib

  • library for visualization with Python
import matplotlib.pyplot as plt
  • other plotting libraries: Seaborn, Plotly, Bokeh, Altair

Anatomy of a figure

[ Find the complete cheatsheet on GitHub in the ‘img/’ folder]

mapping data info into visual info

  • necessary: data (y-axis elements)

  • x-axis elements

  • type of plot: scatter plot, line plot, histogram, bargraph, violin plot, etc.

  • shape, size, and color specification

  • axis ticks and labels

  • legend

  • title

Interfaces/ Styles of Use

  • implicit pyplot interface
    • example: plt.xlabel
    • methods are generally applied to the current axes or figure
  • object-oriented interface
    • example: (ax.set_xlabel)
    • uses methods on a figure or axes object to create other artists, and
    • allows to build a visualization step by step

Recap

Specify a function

import numpy as np
def func(x, a, b, c):
    return a * np.exp(-b * x) + c

x = np.arange(0,10,1)
y = func(x, 1,0.3,0.1)
plt.plot(x, y, 'o')
plt.show()

Fit function

from scipy import optimize
popt, pcov = optimize.curve_fit(func, x, y, p0=(1,1,1))
x_fit = np.arange(0,10,0.1)
plt.plot(x, y, 'o')
plt.plot(x_fit, func(x_fit, *popt))
plt.show()
print(popt)

[1.  0.3 0.1]

Convolve and Deconvolve

  • Signals can be distorted
  • Multiplying something to signal with convolve
  • Remove known distortion from signal deconvolve
  • Where do you see use cases?

Signal analysis (power)

time = np.linspace(start=0, stop=0.5, num=2000)
mu, sigma = 0.25, 0.01
sinewave = np.sin(time * 250 * np.pi)
gaussian = (1 / (np.sqrt(2 * np.pi * np.square(sigma))) *
            np.exp(-(np.square(time - mu) /np.square(2 * sigma))))
ripple = gaussian * sinewave

Signal analysis (power)

np.random.seed(0)
trace = ripple + gaussian*6 + np.random.normal(0, 3, size=time.shape)
plt.plot(time, trace)
plt.show()

Signal analysis (power)

  • Module signal
  • Power spectrum to analyse frequencies
  • e.g. welch

Power spectrum

from scipy import signal
f, Pxx_den = signal.welch(trace, fs=1/time[1], nperseg=1024, average='median')
plt.semilogy(f, Pxx_den)
plt.vlines(125, Pxx_den.min(), Pxx_den.max(),color='r', linestyle='--')
plt.xlabel('Frequency [Hz]')
plt.ylabel('PSD [V**2/Hz]')
plt.title('Power Spectral Density')
plt.show()
  • Returns frequencies f and power densities Pxx_den

Power spectrum

Text(0.5, 0, 'Frequency [Hz]')
Text(0, 0.5, 'PSD [V**2/Hz]')
Text(0.5, 1.0, 'Power Spectral Density')

Filtering data

  • Filtering data is used to remove different components from signal
  • Lowpass, highpass, bandpass
  • Different types of filters (Butterworth, Chebyshev, FIR, etc.)

Filter construction

  • Need different parts
    • Nyquist frequency
    • Filter frequencies (0 to 1 - frequency/Nyquist)
    • Type of filter
    • Order (‘strength’)
  • Outputs are filter coefficients
sampling_rate = 1/time[1]
nyquist = sampling_rate / 2
b, a = signal.butter(N=5, Wn=[100/nyquist, 250/nyquist], btype='band')

Filter construction

w, h = signal.freqz(b, a, fs=sampling_rate, worN=2000)
plt.plot(w, abs(h))
plt.title('Filter frequency response')
plt.xlabel('Frequency [Hz]')
plt.ylabel('Gain')
plt.grid()
plt.axvline(100, color='r', linestyle='--')
plt.axvline(250, color='r', linestyle='--')
plt.show()

Filter construction

Text(0.5, 1.0, 'Filter frequency response')
Text(0.5, 0, 'Frequency [Hz]')
Text(0, 0.5, 'Gain')

Apply filter

  • Different ways of filtering (IIR/FIR)
  • One way or two ways (filtfilt)
ripple_filtered_trace = signal.filtfilt(b, a, trace)
plt.plot(time, ripple_filtered_trace)
plt.plot(time, ripple, '--')
plt.show()

Filter for lowpass

b, a = signal.butter(5, 80/nyquist, btype='low')
lowpass_filtered_trace = signal.filtfilt(b, a, trace)
plt.plot(time, trace)
plt.plot(time, lowpass_filtered_trace, '--')
plt.show()